Solving cubic equations in general

The method I am describing today was given by Girolamo Cardano-an Italian mathematician,who first showed this method in his book Ars Magna.

Let us attempt to solve the general cubic equation:

Later,I will show you how this method is used for some cubic equations.

The first step of solving the equation is called "Depressing the cubic equation".Its purpose is to use a handy substitution to bring the equation down to a much simpler form.

The substitution we are using is

which gives us the equation:

When we multiply everything out we get the equation:

which doesn't look any easier or healthier to solve than the starting equation,but if you look a little closer you can see that the equation is of the form:

which we obtain by moving the free term to the RHS.

The next method was discovered by a mathematician called Scipione dal Ferro.

He said that the equation will be solved if we find such numbers s and t so that the next two equations hold:

He also said that one solution to the cubic (3) is s-t.You can check this by plugging in the values:

But this is still a problem.We don't know how to solve the system for s and t.How are we closer to the answer? Well,we can just express s in terms of t from the equation (4) and just plug it in into the equation (5).

Let's see what we get:

Expanding the equation we obtain a nice "triquadratic" equation, analogous to the biquadratic equation of the fourth degree.Here it is:

Using the substitution

we get:

which is a quadratic equation easily solvable by the quadratic formula.

We can now back substitute this into (4) to get s.From this we can get y=s-t.Then we back substitute y into (2) from which we get x.We now have at least one solution to the starting cubic equation.We can get other roots by polynomial division and quadratic formula.

Here lies the reader who will never open this book. He is forever dead.Taking a new step, uttering a new word, is what people fear most. ― Fyodor Dostoyevsky, Crime and PunishmentThe knowledge of some things as a function of age is a delta function.

Re: Solving cubic equations in general

Here lies the reader who will never open this book. He is forever dead.Taking a new step, uttering a new word, is what people fear most. ― Fyodor Dostoyevsky, Crime and PunishmentThe knowledge of some things as a function of age is a delta function.

Re: Solving cubic equations in general

Here are some example equations (some of them are simpler if you do not use this method, but they are a nice practice):

Here lies the reader who will never open this book. He is forever dead.Taking a new step, uttering a new word, is what people fear most. ― Fyodor Dostoyevsky, Crime and PunishmentThe knowledge of some things as a function of age is a delta function.